You've got
International cricket (Tests/ODIs). Essentially fully professional, though West Indians and Zimbabweans may dispute that.
First class cricket (the one-day format is known as List A by statisticians): played by sides representing a region (so 18 major counties in England: Surrey, Hampshire, Worcestershire etc., territories in Australia: New South Wales, Western Australia, South Australia, Victoria, Queensland and Tasmania, and provinces in South Africa, etc.). It's professional in England and Australia, and semi-professional in most other countries.
Club cricket: Third (and below) divisions, largely amateur for people who just want to play the game in a reasonably organised fashion. The top level of club cricket is usually pretty competitive, though, but leagues are local (there's 25 top level leagues in England, about 10 in Australia, since travel distances have to be kept down). There's the occasional professional player in these leagues as well.